~~

- #Le-- PdiJ 3r'&#v " /' NASA Contractor Report 179538 N- 3 75-2 -

a Sensitivity Analysis and Approximation Methods for General Eigenvalue Problems

(NASA-CE-179538) SENSITIVITY ANALYSIS AND N87-12021 A P P R O X I M A T I O W I I E T H O D S POI? G Z E E F A L E I G B N V A I U Z FEOELEMS Fiual E f p o r t ( V i r q i r i a P o l y t e c h n i c Inst. dna S t a t e Uriv.) 124 f CSCL 20K U fi c la s

G3/39 44923

Durbha V. Murthy and Raphael T. Haftka Virginia Polytechnic Institute and State University Blacksburg, Virginia

October 1986

Prepared for Lewis Research Center Under Grant NAG3-347

NASA National Aeronautlcs and Space Admlnlstratlon

https://ntrs.nasa.gov/search.jsp?R=19870002588 2020-04-07T11:36:24+00:00Z

.

Table of Contents

introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Eigenvalue and Eigenvector Derivatives and their Applications . . . . . . . . . . . . . . . . . 1

1.2 Importance of higher order derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 General Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Approximate Updates to Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.5 Objectives of the Present Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Derivatives of Eigenvalues and Eigenvectors for General Matrices . . . . . . . . . . . . . . . . . . . 9

2.1 Problem Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Normalization of eigenvectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3 Methods of Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.1 Adjoint Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Direct Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.3.3 Modification of Direct Method for Banded Matrices . . . . . . . . . . . . . . . . . . . . . . 26

2.3.4 Iterative Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

I

Efficiency Considerations in Calculating Derivatives ............................. 31

3.1 Operation Counts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3.2 CPU Time Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Calculation of First derivatives of Eigenvalues only . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.4 Calculation of First derivatives of Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . 39

. 3.5 Calculation of First and Second derivatives of Eigenvalues only . . . . . . . . . . . . . . . . 42

Approximate Eigenvalues of Modified Systems ................................. 45

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2 Derivative Based Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.1 Linear Approximation(L1N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.2.2 Quadratic Approximation(QUAD) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2.3 Conservative Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.4 Generalized Inverse Power Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2.5 Generalized Hybrid Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.2.6 Reduction Method(RDN) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Rayleigh Quotient Based Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.3.1 Rayleigh Quotient with Nominal Eigenvectors(RAL1) . . . . . . . . . . . . . . . . . . . . . 55

4.3.2 Rayleigh Quotient with Linearly Approximated Eigenvectors(RAL2) . . . . . . . . . . 56

4.3.3 Rayleigh Quotient with Perturbed Eigenvectors(RAL3) . . . . . . . . . . . . . . . . . . . . 56

4.3.4 Rayleigh Quotient with One-step Inverse Iteration(RAL4) . . . . . . . . . . . . . . . . . 60

4.4 Trace-Theorem Based Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.4.1 One-step Newton-Raphson Iteration(NRT1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

. . . . . . . . . . . . . . . . . . . . . . 4.4.2 Refined One-step Newton-Raphson Iteration(NRT2) 62

4.4.3 One-step Laguerre Iteration(L1T) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5 Other Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.5.1 [I. 11 Pade Approximation(PAD1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

ii

Accuracy and Efficiency of Eigenvalue Approximations .......................... 68

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.2 Accuracy Considerations in Approximating Eigenvalues . . . . . . . . . . . . . . . . . . . . . 69

5.2.1 Order of an Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2.2 First Order Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.2.3 Second Order Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.2.4 Third Order Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2.5 Higher Order Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.2.6 Validation of the Theoretical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.3 Efficiency Considerations in Approximating Eigenvalues . . . . . . . . . . . . . . . . . . . . . 93

5.4 Discussion of Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y6

5.4.1 Case When No Derivatives are Available . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

5.4.2 Case When Derivatives are Available . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

References .......................................................... 108

... Ill

List of Symbols

a

a1

A

B

C

ckjcr

C

dkja

i

Subscript indicating approximate value

Subscript indicating a first approximation

Complex general system matrix of order n

rn-th row of A with the rn-th column deleted

[A - I -U(k) ln~- th column deleted

[cT] - lam

Arbitrary constant

Contribution of the j-th right eigenvector to u!:;

(* - l”i)m-th row and column deleted

Contribution of thej-th left eigenvector to v!$2

Contribution of the j-th right eigenvector to 6 d k )

Contribution of the j- th left eigenvector to 6 d k )

Ratio of p’(h) and p(h)

Identity matrix of order n

Index of eigenvalue or eigenvector

Index of eigenvalue or eigenvector

V

Chapter 1

Introduction

7.7 Eigenvalue and Eigenvector Derivatives and their

Applications

Dynamic response and loads are an important consideration in the

understanding and design of many physical systems. The analytical models

for a wide range of these systems are governed by linear differential equations

so that dynamic model analysis often consists of the solution of an eigenvalue

problem. The eigenvalues and the eigenvectors of the system are fundamental

quantities employed in determining the behavior of the system. Variations in

system parameters lead to changes in the eigenvalues and the eigenvectors

and hence in the response characteristics of the system. It is important to

1

know the magnitude of these variations, and this information is contained in

the derivatives of the system eigenvalues and eigenvectors. Thus derivatives

of eigenvalues and eigenvectors are of immense interest in several fields of

physical sciences and engineering and much research effort has been

expended in developing methods to calculate them.

The applications of these derivatives (or synonymously, sensitivities) are

manifold. Probably the most important applications are in the area of design

optimization. System response sensitivities provide vital information in an

optimization procedure and in general the cost of calculating derivatives is the

dominant contributor to the totai cost in an optimization procedure so that the

efficient computation of eigenvalue and eigenvector derivatives is desirable.

Derivatives can also be effectively used to approximate the eigenvalues and

eigenvectors of a modified system and thus reduce the cost of reanalysis,

substantially lowering the computational burden in optimization tasks. The

derivatives are very useful even in non-automated design procedures becaust:

it is often not clear, from analysis alone, how to modify a design to improve

or maintain the desirable properties. The derivatives identify design

parameters that have the most o r the least influence on the design process

and thus ease the effort in design trend studies.

Derivatives of eigenvalues and eigenvectors are particularly valuable in

calculating the statistics of eigenvalue locations in stochastic dynamic

systems. All physical systems are essentially subject to random environments

and the effect of randomly changing environments is crucial for such systems

2

as missiles, spacecraft, airplanes, land vehicles, buildings and machinery. In

addition, many system models do not have well-defined properties and it is

frequently difficult to predict these properties (for example, stiffnesses)

accurately[l-41. The uncertainties in the system eigenvalues and eigenvectors

are calculated from the estimated uncertainties in the properties of the system

and the environment by using the derivatives of eigenvalues and eigenvectors.

The application of derivatives is not restricted to design-oriented activities.

Sensitivity analysis is also playing an increasing role in determining the

analytical model itself In the areas of system identificatinn and analytical

model improvement using test results, sensitivity analysis is of growing

importance. Much recent work in these fields is directly dependent on the

calculation of eigensystem derivatives.

I .2 Importance of higher order derivatives

While in the past attention was mostly restricted to first order derivatives

of eigenvalues, higher order derivatives are assuming a greater importance

recently. It has been found in certain cases that second order derivatives are

very effective in improving accuracy of approximations[5-13] and efficiency of

design[7,11,12]. In almost all instances, eigenvalues are non-linear functions

of system parameters and a second order approximation offers a wider range

of applicability compared to the first order approximation. Intermediate

variables which may improve the quality of first order approximations are not

genera I I y ava i I ab I e for e i g e nva I u e appro x i mat i o n s . A I so, so me o p t i mi za t i o n

algorithms require second order derivatives, and first order derivatives of

optimal solutions require second order derivatives of constraints[l3]. The use

of second derivatives can also greatly reduce the number of reanalyses

required for the convergence of an optimization procedure[ll,14]. Further, in

certain optimization algorithms, second order approximations for eigenvalue

constraints can drastically relax the move limits, thus achieving a nearly

optimum trajectory, and can virtually eliminate the need for trial and error

adjustment of move limits, thus improving the performance of the

optimizer[l4]. Looking at another aspect, in problems where instabilities are

to be avoided, a first order calculation may completely fail to detect

instabilities[6]. References [15,16] also offer examples of the usefulness of

second order derivatives.

1.3 General Matrices

The problem of calculating the derivatives of symmetric and hermitian

eigenproblems is relatively simple and solution procedures are

well-established, e.g.[17-211. However, many physical problems give rise to

non-self-adjoint formulations and thus lead to general matrices. An important

example is aeroelastic stability which requires the solution of eigenproblems

4

with complex, general and fully populated matrices. General matrices are also

obtained in damped structural systems and in network analysis and control

system design where the eigenvalues are usually called poles. In the present

study, the emphasis is on general matrices and the special properties of

matrices, such as symmetry, are not considered.

7.4 Approximate Updates to Eigenvalues

Eigenvalue calculation for any but the smallest systems is an expensive

process and is a major contributor to the computational expense of the typical

dynamic analysis. In the design of structural systems, an iterative

desigdanalysis process is performed until a satisfactory design is achieved.

The cycle of the iterative process consists of an update of the structural

design, a response analysis and calculation of updated responses and loads.

When the process is not automated, a revision of the mathernatical model may

also be present. In typical dynamic analyses, each design cycle is an

expensive process because the mathematical models are large and the time

and effort required for analysis of an updated design are often prohibitive and

greatly reduce the effectiveness of the design process. Apart from the

computational effort, organizational effort can also be substantial.

Appreciable cost savings can be realized if a quick evaluation of a change

in system response resulting from the design changes is possible. An

5

efficient, even i f approximate, evaluation of eigenvalues of a modified system

is valuable in these applications. The value of an approximation depends on

its efficiency as well as its accuracy in applications. A quick approximation

that is valid in a very limited range of design space is of little use as it can

severely reduce the global efficiency by requiring many more evaluations for

convergence of the design process.

Several researchers have worked on suitable approximations for

eigenvalues of a modified design. However, in the past, attention seems to

have been restricted to real symmetric systems which have eigenvalues in the

real number field. It is one of the objectives of the present work to extend the

techniques of approximation to general (complex non-hermitian) systems and

perform a comparative analysis of the various techniques.

1.5 Objectives of the Present Work

The objectives of the present research are to:

1. Review and perform a coniparative analysis of the various methods

available for calculating the derivatives of eigenvalues and eigenvectors

of general matrices. c

2. Propose and evaluate some modifications to existing techniques.

6

. k

I

m

n

P

Pa

P ( V

P

Q

r

R

S

S

t

Index of eigenvalue or eigenvector

Number of eigenvalues of interest

Number of design variables

Number of combinations of rn objects taken two at a time

Order of matrices A and A,

Vector of design variables

a-th design variable

Characteristic equation of A

Matrix o i order 2 defined in eq. (4.2.9j

Matrix of order 2 defined in eq. (4.2.9)

Right hand side in Direct method

Generalized Rayleigh Quotient

Order of Approximation

Right hand side in Direct method

'm-th row deleted

Sum of the elements on the principal diagonal

Superscript to denote the transpose

k-th right eigenvector of matrix A

rn-th element of dk)

Matrix whose columns are the right eigenvectors

k-th left eigenvector of matrix A

rn-th element of dk)

Matrix whose columns are the left eigenvectors

vi

X "m-th row deleted

Y

Y1

y2

Z

a

A

K

h

0

" m t h row deleted

Defined in eq. (2.3.28)

Defined in eq. (2.3.32)

Defined in eq. (5.2.11)

Index of design variable

Subscript indicating derivative with respect to pa

Subscript indicating second derivative with respect to ptx and p~

Subscript indicating derivative with respect to pp

Prefix indicating estimated change

Prefix indicating actual change

Value of design variable lying between p,r and pa + Aptx

Measure of the sparsity of A,cx or AA

Eigenvalue of A

Eigenvalue of A,

Diagonal matrix of eigenvalues

Superscript to denote the conjugate transpose

Subscript indicating nominal value

Chapter 1

Introduction

7.7 Eigenvalue and Eigenvector Derivatives and their

Applications

Dynamic response and loads are an important consideration in the

understanding and design of many physical systems. The analytical models

for a wide range of these systems are governed by linear differential equations

so that dynamic model analysis often consists of the solution of an eigenvalue

problem. The eigenvalues and the eigenvectors of the system are fundamental

quantities employed in determining the behavior of the system. Variations in

system parameters lead to changes in the eigenvalues and the eigenvectors

and hence in the response characteristics of the system. It is important to

1

know the magnitude of these variations, and this information is contained in

the derivatives of the system eigenvalues and eigenvectors. Thus derivatives

of eigenvalues and eigenvectors are of immense interest in several fields of

physical sciences and engineering and much research effort has been

expended in developing methods to calculate them.

The applications of these derivatives (or synonymously, sensitivities) are

manifold. Probably the most important applications are in the area of design

optimization. System response sensitivities provide vital information in an

optimization procedure and in general the cost of calculating derivatives is the

dominant contributor to the total cost in an optimization procedure so that the

efficient computation of eigenvalue and eigenvector derivatives is desirable.

Derivatives can also be effectively used to approximate the eigenvalues and

eigenvectors of a modified system and thus reduce the cost of reanalysis,

substantially lowering the computational burden in optimization tasks. The

derivatives are very useful even in non-automated design procedures because

it is often not clear, from analysis alone, how to modify a design to improve

or maintain the desirable properties. The derivatives identify design

parameters that have the most or the least influence on the design process

and thus ease the effort in design trend studies.

Derivatives of eigenvalues and eigenvectors are particularly valuable in

calculating the statistics of eigenvalue locations in stochastic dynamic

systems. All physical systems are essentially subject to random environments

and the effect of randomly changing environments is crucial for such systems

2

, as missiles, spacecraft, airplanes, land vehicles, buildings and machinery. In

addition, many system models do not have well-defined properties and it is

frequently difficult to predict these properties (for example, stiffnesses)

accurately[l-41. The uncertainties in the system eigenvalues and eigenvectors

are calculated from the estimated uncertainties in the properties of the system

and the environment by using the derivatives of eigenvalues and eigenvectors.

The application of derivatives is not restricted to design-oriented activities.

Sensitivity analysis is also playing an increasing role in determining the

analytical mndel itself In t h e areas of system identification and analytical

model improvement using test results, sensitivity analysis is of growing

importance. Much recent work in these fields is directly dependent on the

calculation of ei gensyst em derivatives.

7.2 Importance of higher order derivatives

While in the past attention was mostly restricted to first order derivatives

of eigenvalues, higher order derivatives are assuming a greater importance

recently. It has been found in certain cases that second order derivatives are

very effective in improving accuracy of approximations[5-13] and efficiency of

design[7,11,12]. In almost all instances, eigenvalues are non-linear functions

of system parameters and a second order approximation offers a wider range

of applicability compared to the first order approximation. Intermediate

3

variables which may improve the quality of first order approximations are not

generally available for eigenvalue approximations. Also, some Optimization

algorithms require second order derivatives, and first order derivatives of

optimal solutions require second order derivatives of constraints[l3]. The use

of second derivatives can also greatly reduce the number of reanalyses

required for the convergence of an optimization procedure[ll,14]. Further, in

certain optimization algorithms, second order approximations for eigenvalue

constraints can drastically relax the move limits, thus achieving a nearly

optimum trajectory, and can virtually eliminate the need for trial and error

adjustment of move limits, thus improving the performance of the

optimizer[l4]. Looking at another aspect, in problems where instabilities are

to be avoided, a first order calculation may completely fail to detect

instabilities[6]. References [15,16] also offer examples of the usefulness of

second order derivatives.

1.3 General Matrices

The problem of calculating the derivatives of symmetric and hermitian

eigenproblems is relatively simple and solution procedures are

well-established, e.g.[17-21). However, many physical problems give rise to

non-self-adjoint formulations and thus lead to general matrices. An important

example is aeroelastic stability which requires the solution of eigenproblems

4

with complex, general and fully populated matrices. General matrices are also

obtained in damped structural systems and in network analysis and control

system design where the eigenvalues are usually called poles. In the present

study, the emphasis is on general matrices and the special properties of

matrices, such as symmetry, are not considered.

7.4 Approximate Updates to Eigenvalues

Eigenvalue calculation for any but the smallest systems is an expensive

process and is a major contributor to the computational expense of the typical

dynamic analysis. In the design of structural systems, an iterative

desigdanalysis process is performed until a satisfactory design is achieved.

The cycle of the iterative process consists of an update of the structural

design, a response analysis and calculation of updated responses and loads.

When the process is not automated, a revision of the matherriatical model may

also be present. In typical dynamic analyses, each design cycle is an

expensive process because the mathematical models are large and the time

and effort required for analysis of an updated design are often prohibitive and

greatly reduce the effectiveness of the design process. Apart from the

computational effort, organizational effort can also be substantial.

Appreciable cost savings can be realized if a quick evaluation of a change

in system response resulting from the design changes is possible. An

5

efficient, even if approximate, evaluation of eigenvalues of a modified system

is valuable in these applications. The value of an approximation depends on

its efficiency as well as its accuracy in applications. A quick approximation

that is valid in a very limited range of design space is of little use as it can

severely reduce the global efficiency by requiring many more evaluations for

convergence of the design process.

Several researchers have worked on suitable approximations for

eigenvalues of a modified design. However, in the past, attention seems to

have been restricted to real symmetric systems which have eigenvalues in the

real number field. It is one of the objectives of the present work to extend the

techniques of approximation to general (complex non-hermitian) systems and

perform a comparative analysis of the various techniques.

1.5 Objectives of the Present Work

The objectives of the present research are to:

1. Review and perform a comparative analysis of the various methods

available for calculating the derivatives of eigenvalues and eigenvectors

of general matrices. c

2. Propose and evaluate some modifications to existing techniques.

6

3. Formulate guidelines for selecting the most efficient computational

algorithm for particular applications.

4. Review the various approximations to eigenvalues for real symmetric

systems and extend them to the case of complex general systems.

5. Compare the various approximations in terms of efficiency and accuracy

for some systems.

Outline

Chapter 2 reviews the various methods available for the calculation of

derivatives of eigenvalues and eigenvectors for general matrices. The

important consideration of normalization of eigenvectors of complex general

matrices, which has not been adequately dealt with in the literature, is

discussed. A new algorithm to calculate the derivatives of eigenvalues and

eigenvectors simultaneously, based on a better normalizing condition, is

described and important numerical aspects regarding the implementation of

the algorithm are discussed, with consideration being given to sparse

matrices. The various algorithms are classified as Adjoint or Direct.

The efficiency considerations of the various algorithms are examined in

Chapter 3. Operation counts are presented in terms of matrix size, number

of design parameters, and the number of eigenvalues and eigenvectors of

7

interest. Actual CPU times are also presented for typical matrices for a range

of parameters that influence the efficiency of the algorithms.

Chapter 4 provides a survey of approximation methods proposed in the

literature for real symmetric matrices, describing their special features. The

approximation methods are extended to complex general matrices wherever

feasible. Some approximation methods which do not seem to have been

applied in the past are also presented. The approximation methods are

classified on the basis of their theoretical origin.

Numerical results from applying the proposed techniques of

approximations are presented in Chapter 5. Operation counts are presented

in terms of matrix size, number of design parameters, number of eigenvalues

of interest and the number of times the approximation is to be performed. The

approximation techniques are applied to typical matrices and random matrices

and are evaluated in terms of their accuracy and efficiency.

Chapter 6 contains the concluding remarks. General guidelines for

selecting approximation methods a n d algorithms for calculation of eigenvalue

and eigenvector derivatives are summarized. This chapter also contains

remarks about the limitations of the present work and recommendations for

further research.

8

Chapter 2

Derivatives of Eigenvalues and Eigenvectors for

General Matrices

2.1 Problem Definition

The matrix eigenproblem is defined as follows:

Au(k) = h(k)u(k)

and the corresponding adjoint problem is

v ( ~ ) ~ A = ( k ) v (k)T

9

(2.1 .l)

(2.1.2)

where A is a general complex matrix of order n and dk) and dk) are the

k -th eigenvalue and right and left eigenvectors respectively. A superscript T

denotes the transpose.

(The adjoint problem is defined by some authors in an alternative form as

where superscript * denotes a conjugate-transpose. However, the notation of

eq. (2.1.2) is more popular in the literature on structural dynamics).

The eigenvalues and eigenvectors are complex and do not necessarily

occur in complex-conjugate pairs. Al l eigenvalues are assumed to be distinct.

The matrix A and hence, and dk) are functions of design

parameter vector p with individual parameters denoted by Greek subscripts,

e.g. pa. Derivatives with respect to ptx are denoted by the subscript , (1 e.g.,

- _ ;A - A , a . All the design variables are assumed to be real. ?P(X

The well-known biorthogonality properties of the eigenvectors are given

(2.1.3)

and

(2.1.4)

10

Note that, the left hand side of eq. (2.1.3) is not an inner product as usually

understood, since di) and/or uu) may be complex vectors. Note also that the

left eigenvectors of A are the right eigenvectors of AT and vice versa.

2.2 Normalization of eigenvectors

The eigenvectors dk) and dk) are not completely defined by eqs. (2.1.1)

and (2.1.2). A normalization condition has to be imposed to obtain unique

eigenvectors. For brevity, let us consider only the normalization of the right

eigenvector. A normalizing condition frequently imposed in the self-adjoint

case is the following:

(2.2.1)

However, it is not always possible to use eq. (2.2.1) for non-self-adjoint

can equal zero or a very small number causing numerical problems as

difficulties. This is true even if the matrix A is real, as shown by the example r o -11

matrix A = . Unfortunately, considerable confusion exists in the

literature regarding this point and several authors arbitrarily adopted

eq.(2.2.1) as a normalizing condition for non-self-adjoint problems,

e.g.[ll,12,15,22-25]. In this respect, the formulations of these references are

not rigorous for general matrices.

One possible way to avoid the above difficulty is to replace eq.(2.2.1) by

11

(2 2.2)

where superscript

the difficulties of

* denotes a conjugate-transpose. Eq. (2.2.2) is not prone to

eq. (2.2.1) because u ( ~ ) ' u ( ~ ) is always guaranteed to be

non-zero. But, eq.(2.2.2) is not a complete normalizing condition as it does not

render the eigenvector unique. If u satisfies eq.(2.2.2), then w = ueic, where

i = J-1 and c is an arbitrary real number, also satisfies eq.(2.2.2). Despite

this limitation, eq.(2.2.2) can be used satisfactorily in certain

formulations[26,27].

Another normalization condition, inspired by the biorthogonality property

of the left and right eigenvectors, is

Eq.(2,2.3) also does not render the eigenvectors unique. since a pair

dk), dk) can be replaced by cu(k),(llc)v(k), where c is an arbitrary non-zero

complex number, and still satisfy eq. (2.2.3). Again, this is not necessarily a

severe restriction for calculation of the derivatives of eigenvectors[5,16,28-301.

It must, however, be emphasized that if the eigenvector is not unique, nor is

its derivative.

The norma I i zat i o n condition

(2.2.4)

is very attractive because it renders the eigenvectors unique and at the same

time, the index m can be chosen easily to avoid ill-conditioning. Apparently,

only Nelson[31] used this normalizing condition in obtaining the derivatives

of eigenvectors. The index, m , may be chosen such that

Another choice for m, used by Nelson[31], is

(2.2 -5)

(2.2.6)

The nature of uncertainty of the derivative of the eigenvector is of some

interest. Without a normalizing condition, an eigenvector is uncertain to the

extent of a non-zero constant multiplier. The derivative of an eigenvector is

uncertain to the extent of an additive multiple of that-eigenvector. To show

this, let dk) be an eigenvector so that w ( ~ ) = C U ( ~ ) is also an eigenvector.

Then, if p a is a design parameter,

(2.2.7)

The quantity d = - is not zero since the quantity c is not really a apt1

constant, but is a function of the nature of the normalization criterion. In

practice, the constant d depends on the way the eigenvectors dk) and w ( ~ ) are

normalized.

13

2.3 Methods of Calculation

The various methods of calculating the derivatives of eigenvalues and

eigenvectors can be divided into three categories:

1. Adjoint Methods, which use both the right and the left eigenvectors.

2. Direct Methods, which use only the right eigenvectors.

3. iterative Methods, which use an iterative algorithm that converges to the

required derivatives.

2.3.1 Adjoint Methods

The first expressions for the derivatives of eigenvalues of a general matrix

seem to have been derived by Lancaster[32]. Considering only a single

parameter, Lancaster obtained the following expressions for the first and

second derivatives of an eigenvalue:

(2.3.1)

j+k

Eq. (2.3.1) can be obtained in the following manner. Differentiate eq. (2.1 . l )

with respect to the parameter pa to obtain

( k ) = ( k ) (4 + h(kIu!k,) A ,u(~) + Au, a A, flu (2.3.3)

The last terms in the expressions on both sides are equal due to eq.

(2.1.2), so that

Eq. (2.3.1) follows immediately. Eq. (2.3.2) is derived in its more general form

later. An expression corresponding to eq. (2.3.1) for a non-linear eigenvalue

pro blem

A(A.)u(~) = 0 (2.3.6)

was obtained by Pedersen and SeyranianI331 in a similar manner as

15

(2.3.7)

To obtain the second derivatives of eigenvalues, the first derivatives of left

and right eigenvectors are calculated either explicitly[5,12,16,26,30] as in eq.

(2.3.16) o r irnplicitly[12,15,32] as in eq. (2.3.18). Since the eigenvalues are

assumed to be distinct, the set of eigenvectors forms a basis for the n-space

and the first derivatives of eigenvectors can be expressed in terms of the

eigenvectors as

(2.3.8)

Now, the calculation of the first derivatives of eigenvectors reduces to the

evaluation of the coefficients ckjIr and dkjI,.

Premultiplying eq. (2.3.3) by & I T , where j + k , we get

Now, substituting the expansions of eq. (2.3.8) and using eqs. (2.1 .l) and (2.1.2)

and the bi-orthogonality property of eq. (2.1.3), we obtain

16

(2.3.10)

Proceeding in a similar manner after differentiating eq. (2.1.2) with respect to

the parameter pa, we obtain

(2.3.1 1)

The above expressions for the coefficients ck ja and dkj(r were obtained by

Rogers [ 291.

It can be observed that

(2.3.12)

Reddy[34] derived an equivalent expression for the response derivative by

casting the derivative as the solution of a forced response problem for the

same system.

Note that, in view of eq. (2.2.7), the coefficients c ~ ~ ( ~ and d k k a in eq. (2.3.8)

are arbitrary and depend on the normalization of the eigenvectors. For

example, if eq. (2.2.4) is used to normalize the right eigenvectors, then

and if eq. (2.2.3) is used to normalized the left eigenvectors, then

- d k k a - - C k k a

(2.3.13)

(2.3.14)

17

It has been proposed[31,35,36] that the eigenvector derivative be

approximated by using less than the full set of eigenvectors in the expansion

of eq. (2.3.8) so that the evaluation of eigenvector derivative by Adjoint method

could become cheaper. This variant of Adjoint method has received mixed

reports in the literature[31,35]. The quality of such an approximation is difficult

to assess beforehand and the selection of the number of eigenvectors to be

retained in the expansion is problem dependent. It is not considered in this

work because a meaningful comparison with other methods cannot be easily

be made. However, this consideration should not be ignored while

implementing the sensitivity calculations for particular problems.

The expressions for the second derivatives of eigenvalues were obtained

by Plaut and Huseyin[30]. For the sake of simplicity in expressions, let us

assume, without loss of generality, that the right and left eigenvectors are

normalized as in eq. (2.2.3). Eq. (2.3.1) can then be written as

(2.3.15)

Differentiating with respect to a parameter pp uncorrelated to the parameter

pa, we obtain

(2.3.16)

which can be equivalently written, without involving the derivative of the left

eigenvector, as

18

Eq. (2.3.16) can be rewritten using eqs. (2.3.10) and (2.3.11) as

(2.3.1 8)

j f k

Crossley and Porter[5,28] derived similar expressions for derivatives with

resi;ect to t h e z lcm~f i ts of t h C matrix. Expr~ss ion for t h ~ I?'-th order diagonal

derivative was derived by Elrazaz and Sinha[9] and it is

(~h(k) TN-1 ( k )

dP, dp, N - l a u ... - N

(2.3.19)

Morgan[37] developed a different computational approach for the

derivative of an eigenvalue without requiring the eigenvectors explicitly. His

expression is equivalent to

19

trace of {[adj(A - h(k)l)]A .} 1 (k) = ’b, a

trace of adj(A - l ~ ( ~ ) l ) (2.3.20)

The corresponding expression for derivatives with respect to matrix elements

was derived by Nicholson[38].

It can however be shown that[39]

where tk is a constant and that[40]

trace of adj(A -

(2.3.21)

(2.3.22)

Thus, in the computation of adj(A - h(k)i), both right and left eigenvectors

are implicitly computed, in view of eq. (2.3.21). Eqs. (2.3.22) also show that

Morgan’s eq. (2.3.20) is equivalent to Lancaster’s eq. (2.3.1). Woodcock[41]

also obtained formulas involving the adjoint matrix for the first and second

derivatives of eigenvalues. An operation count shows that calculation of the

adjoint matrix is several times more expensive than the explicit calculation of

right and left eigenvectors so that Lancaster’s formula is preferable to

formulas requiring the adjoint matrix. This conclusion is also supported by

sample computations[42]. In addition, although eq. (2.3.20) was used

satisfactorily for small problems[43,44], numerical difficulties were reported for

reasonably large problems[45]. Woodcock’s formula for the second derivative

20

of the eigenvalue requires a partial derivative of the adjoint matrix and this is

so complicated that Woodcock himself recommends the finite difference

method. Formulas due to Morgan and Woodcock are not therefore considered

in the following.

In calculating the derivatives by Adjoint Methods, i.e., using eqs. (2.3.1),

(2.3.8)-(2.3.18),

the first derivative of an eigenvalue requires the corresponding right and

left eigenvectors.

the first derivative of an eigenvector requires all the left and right

eigenvectors.

the second derivative of an eigenvalue requires the corresponding right

and left eigenvectors and their first derivatives.

2.3.2 Direct Methods

The second category comprises methods that evaluate the derivatives

using only the right eigenproblem. Direct methods typically involve either the

evaluation of the characteristic polynomial or the solution of a system of linear

simultaneous equations without requiring all the left and right eigenvectors.

Methods requiring the evaluation of the characteristic polynomial and the

derivative of the determinant[45,46] are O(n5) processes while other methods

21

considered here are at most O(n3) processes. In addition, the determination

of the characteristic polynomial is, in general, an unsatisfactory process with

respect to numerical stability, even when all the eigenvalues are

well-conditioned[47]. While numerically stable algorithms have been

proposed for evaluation of the characteristic polynomial[48], the computational

expense still seems to be formidable. Hence, we do not consider these

methods. Methods requiring the solution of a system of equations have the

particularly attractive feature that the coefficient matrix needs to be factored

only once for each eigenvalue regardless of the number of parameters and the

order of the derivatives required. Thus, they are very useful in applications

where higher order derivatives are required.

The earliest method in this class is due to Garg[22] who obtained the first

derivatives of the eigenvalue and the eigenvector by solving two systems of

(n + 1) equations each in the real domain, without requiring any left

eigenvectors. However, his formulation involves several matrix

multiplications. Rudisill[23] proposed a scheme in which only the

corresponding left and right eigenvectors are required to calculate the first

derivative of the eigenvalue and the eigenvector. This was refined by Rudisill

and Chu[24] to avoid calculating the left eigenvectors altogether. Solution of

a system of only (n + 1) equations is required (though in the complex domain)

to obtain the first derivatives of eigenvalue as well as eigenvector. Extension

to higher order derivatives is straightforward. Cardani and Mantegazza[Zli]

22

proposed solution methods of the same formulation for sparse matrices and

extended it to the quadratic eigenproblem.

One weakness that is common to all the above formulations that do not

require left eigenvectors[22-25] is that they rely on the normalization condition

given by eq. (2.2.1), which is unreliable as discussed earlier.

Nelson[31] circumvented this difficulty by using the norrnalizing conditions

(2.3.23)

However, ine iorrriuiaiiuii U i RUdisiii and Chu is s u p c r i ~ r !G Nclsa::‘s

formulation in that it does not require any left eigenvectors.

In this work, we propose a variation of the Rudisill and Ctiu formulation

which does not rely on the questionable normalizing condition of eq. (2.2.1)

and at the same time requires no left eigenvectors.

Differentiating eq. (2.1 . l ) , we get

which can be rewritten in partitioned matrix form as

(2.3.24)

(2.3.25)

Now, we impose the normalizing condition of eq. (2.2.4). Differeritiation of

eq. (2.2.4) yields,

23

(2.3 26)

Because of eq. (2.3.26), the rn-th column of the coefficient matrix in eq.

(2.3.25) can be deleted. Eq. (2.3.26) also reduces the number of unknowns by

one so that eq. (2.3.25) is now a system of n equations in 11 unknowns. Eq.

(2.3.25) is rewritten as

By, = r

where

(2.3 2 7 )

r = - A , , U ( ~ ) (2.3.28)

To get second derivatives, differentiate (2.3.24) with respect to pi{ arid yet,

or , in partitioned matrix form,

Following the same reasoning as before, eq. (2.3.30) is written as

By;, = s

where

(2.3.3 1 )

( k ) U Y2 = { -$>

, alj with rn-th element deleted

Note that, if is a distinct eigenvalue of A arid i f us!) f- 0, then the matrix

A is of rank (n - 1) and the rn-th colunin that is deleted is linearly dependent

on the other columns. Hence the matrix E3 is non-singular. The matrix B will

also be well-conditioned if uh!) is the largest component in the eigenvector

dk) and the matrix A is itself not ill-conditioned. The vectors y1 and y2 can be

obtained by standard solution methods. If the matrix A is banded or i f the

derivatives of both right and left eigenvectors are required, i t may be more

efficient to use a partitioning scheme as described below.

25

2.3.3 Modification of Direct Method for Banded Matrices

Equations (2.3.27) and (2.3.31) can be written as

Let

Eq. (2.3.33) is a system of n equations.

be normalized so that u,$) = u,$& = constant

Writing the m-th equation

separately, we have, if the superscript ( k ) is omitted for notational

convenience,

cx,,, - h (,X = t

and

T - a m x , a - 1, aUmO - r m

where

= (A - A')m-th row and column deleted

- - x , a ", n m-th row deleted

= Um-th row deleted

= rm-th row deleted

(2.3.34)

(2.3.35)

T a, = rn-th row of A with the m-th column deleted

From (2.3.35),

From (2.3.34),

x , a = C- ' (h .x + t)

Eliminating x, we have

where

b, = [CT]-'a,

Proceeding in a similar manner for the left eigenvector,

where

- Y , a - " , a m-th row deleted

(2.3.36)

(2.3.37)

(2.3.38)

(2.3.39)

Y = "m-th row deleted

27

= (‘drn-th row deleted

rI being the appropriate right hand side.

Thus the following procedure can be used to obtain the derivatives A, (1 and

1.

2.

3.

4.

5.

Form a LU decomposition of the matrix C .

Solve b, = [CT]- la , by forward substitution.

Calculate h from (2.3.38).

Calculate x from (2.3.37) by backward substitution.

Expand x, to u , ~ setting u ,,,, = 0.

If the derivatives v , ~ of the left eigenvectors are also required, only three

further steps are needed.

6.

7.

8.

Calculate y, a from (2.3.39) by forward substitution.

Expand y, to v setting vn,, = 0.

Normalize v, ( x appropriately depending on the normalization of v. For

example, to obtain the derivative of the left eigenvector that satisfies the

normalization condition of eq. (2.2.3), subtract ( v ~ u , ~ + vTau)v .

28

The matrix C needs to be factored only once. Also, the matrix C retains

the bandedness characteristics of the original matrix A, so that advantage can

be taken of it. Furthermore, higher order derivatives can be obtained by

merely substituting an appropriate right hand side vector, r. However, higher

order derivatives can suffer in accuracy because of accumulated round-off

error.

The conditioning of matrix C needs some comment. Note that C is

obtained from the singular matrix (A - h(k)l) by deleting both the row and

column corresponding to index rn. Hence, for matrix C to be non-singular, one

must make sure that the rn-th row is linearly dependent on the other rows as

well as that the m-th column is linearly dependent on the other columns. In

other words, C is non-singular iff u# Z. 0 and v x ) # 0 . If v# is very small

compared to the largest element in dk), steps 2 and 4 in the above procedure

will give inaccurate results even if u# is the largest element in dk). In

general, it is not possible to make a good choice for m without the knowledge

of the left eigenvector. Since the calculation of left eigenvector using forward

substitution in an inverse iteration scheme is cheap (as explained later in

Section 3.1), it is suggested that the left eigenvector be calculated and the

index m be chosen as in eq.(2.2.6). This is the same criterion used by

Nelson[31] and will assure as well-conditioned a matrix C as possible.

In summary, we note that, in calculating derivatives by Direct Method,

left eigenvectors are not used.

29

a complete solution of the eigenvalue problem is not required, if the

derivatives of only a few of the eigenvalues and eigenvectors are sought.

This is in contrast to Adjoint Method which requires all the left and right

eigenvectors to calculate the first derivative of any eigenvector.

calculation of any derivative requires the solution of a system of linear

equations.

only one matrix factorization needs to be performed for all orders of

derivatives of an eigenvalue and its corresponding right and left

eigenvectors.

2.3.4 Iterative Methods

Andrew[27] proposed an iterative algorithm to calculate the first

derivatives of eigenvalues and eigenvectors. This algorithm is a refined and

generalized version of the iterative scheme developed by Rudisill and Chu[24].

Except for the dominant eigenvalue, the convergence of this algorithm seems

to be very much dependent on the choice of the initial values for the

derivatives. To be efficient for non-hermitian matrices, this iterative method

requires a complex eigenvalue shifting strategy which is not easy to

implement. Hence this method is not considered.

30

Chapter 3

Efficiency Considerations in Calculating Derivatives

In order to establish criteria for the selection of the most efficient algorithm

for calculating the derivatives of eigenvalues and eigenvectors in a given

application, we compare the operation counts and actual CPU times required

by Adjoint and Direct methods. The decision as to which algorithm is best is

necessarily problem-dependent, The comparison is, however, described in

terms of three variables that can usually be ascribed to a given problem and

which significantly influence the decision. These variables are

1. the size of the matrix n

2. the number of design parameters m

3. the number of eigenvalues of interest 1.

31

3.1 Operation Counts

To start with, let us consider the operation counts (multiplications and

divisions only) for the adjoint methods given by eqs. (2.3.1),(2.3.8)-(2.3.18) and

the direct methods given by eqs.(2.3.27)-(2.3.32). They are summarized in

Table 1. It should be noted that the operation counts represent an estimate

of the actual number of operations performed by a solution routine and include

only the most significant terms. The actual number of operations will vary

slightly depending on programming details. The effect of the sparsity of the

matrix derivative A , a is modeled in the operation counts by the parameter K ,

defined such that the number of operations in evaluating the product A .u is

equal to Kn2(that is, K = 1 corresponds to a full A *).

The eigenvalues are calculated using the EISPACK subroutine package[49]

by first reducing the matrix to upper hessenberg form using unitary similarity

transformations and then applying the QR algorithm. The number of

operations and the CPU time for calculating the eigenvalues is not relevant in

evaluating the methods to calculate the derivatives. The operation count for

eigenvalue computation is given only for comparison.

The right eigenvectors are calculated by inverse iteration on the same

upper hessenberg matrix used for calculating the eigenvalues and are back

transformed using standard subroutines in the package EISPACK. The

corresponding operation count is given in Table 1. The inverse iteration

algorithm is an extremely powerful method for computing eigenvectors and is

32

much superior in accuracy as well as speed of convergence to the common

alternative algorithms based on the solution of homogeneous equations or

direct iteration. Algorithms based on the solution of homogeneous equations

are limited in their accuracy by the accuracy of the eigenvalue and those

based on direct iteration are limited in their convergence, particularly for

eigenvectors not corresponding to either the largest or the smallest

e i g e nva I u e.

For the calculation of left eigenvectors, it is important to note that there is

no need to repeat the process with the transposed matrix. The left

eigenvectors are obtained cheaply using forward substitution in place of

backward substitution in the inverse iteration process. There is also no need

to repeat the matrix factorization. A subroutine was written to calculate the

right and left eigenvectors in this manner and the corresponding operation

count is given in Table 1.

Table 1 gives the operation count of evaluating the individual steps. To

obtain the number of operations involved in evaluating the derivatives, we

must add the operation counts for all the steps required in the calculations.

These counts are given in the following discussion.

33

Table 1. Operation Counts

Eigenvalues and Eigenvectors

Ope rat ion Operation Count

Eva1 uat i o n of eig enval ues Eva1 uat ion of right eigenvectors

Evaluation of left eigenvectors

Adjoint Methods

Operat ion

Evaluation of eq. (2.3.1) Evaluation of eq. (2.3.8),

(2.3.10),(2.3.11)

Evaluation of eq. (2.3.18)

Operation Count

I (T) n2K

Direct Methods

Ope rat ion Operation Count

n3 I(-) LU decomposition of matrix B

Formulation and solution of eq.(2.3.27)

Formulation and solution of eq.(2.3.31)

/m3*(K + 1)

I ( Y ) n2(3, + 1)

34

3.2 CPU Time Statistics

In the following tables, computational cost for the calculation of the first

and second derivatives of eigensystems are compared for matrices of order

20, 40 and 60. The CPU time statistics are obtained on the IBM 3084 computer

using the VS-FORTRAN compiler with no compiler optimization. The

correlation between operation counts and CPU times is shown in Tables 2 and

3. The ratio of operation count(0C) and CPU time for various operations,

tabulated in Tables 2 and 3, is about lo5 operations per CPU second with a

variablity of 27 percent.

The typical matrices are generated for the dynamic stability aeroelastic

analysis of a compressor stage rotor with mistuned blades. The geometric and

structural parameters of the rotor and formulation and method of analysis are

the same as those of NASA Test Rotor 12 described in reference[50] except

that the number of blades and the torsional frequencies are varied. The

torsional frequency values are selected randomly from a population of mean

1.0 and standard deviation 0.01. The standard deviations of the actual

samples are slightly different.

35

Table 2. Correlation between Operation Count(0C) and CPU time

Calculation of right eigenvectors

n I OC/CPU seconds (xl 04)

60 60 60 10

8.6 8.3

Calculation of right and left eigenvectors

n

60 60

n

60 60 60 60

I OC/CPU seconds (x104)

60 10

Evaluation of eq. (2.3.1)

8.5 9.2

I m OC/CPU seconds (x104)

60 10 60 5 10 10 10 5

10.7 10.7 10.7 10.7

Evaluation of eq. (2.3.8),(2.3.10),(2.3.11)

n I Ill OC/CPU seconds (x104)

60 60 10 60 60 5 60 10 ' 10 60 10 5

12.0 11.9 11.9 11.9

36

Table 3. Correlation between Operation Count(0C) and CPU tirne(Contd.)

n

60 60

n

60 60

I

60 10

I

60 10

Evaluation of eq. (2.3.18)

m OC/CPU seconds (x104)

10 5

Decomposition of matrix B

9.1 7.4

OC/CPU seconds (xl 04)

8.5 9.2

Evaluation of eq. (2.3.27)

n I m OC/CPU seconds (xl 04)

60 60 10 60 10 5

12.9 13.0

37

3,3 Calculation of First derivatives of Eigenvalues only

Adjoint Method

Direct Method

Operation Count

7 2

/(-n2 -k Kmn2)

n3 I[- + (K + l)mn2] 3

It is clear from the operation count that the Adjoint Method, which is an

O(n2) process, is superior to Direct Method, an O(n3) process, for large n. The

number of design variables and the number of eigenvalues of interest have no

bearing on this conclusion. As the order of the matrix increases, the direct

method becomes more expensive. For example, for 5 design variables and

10 eigenvalues of interest, the CPU time for the Direct method is 2.3 times

more expensive than for the Adjoint Method for n = 20, and for n = 60, the

ratio is 3.0.

3.4 Calculation of First derivatives of Eigenvalues and

Eigenvectors

Operation Count

Adjoint Method

Direct Method

When the derivatives of both eigenvalues and right eigenvectors are

required, the choice of method is dependent on the values of / and m. When

very few eigenvalues are of interest, the Direct method is cheaper. When

many eigenvalues are of interest, the Direct method is more expensive than

the Adjoint method. However, this effect of the number of eigenvalues of

interest is less significant when the number of design variables is large. A s

the number of design variables increases, the direct method becomes more

competitive, even when all eigenvalues are of interest. For a 60 x 60 full

( K = 1) matrix, this is illustrated in Figure 1 on page 40.

The operation count shows that the computation by adjoint method of

eigenvector derivative, which is necessary for the second derivative of

eigenvalue, is an O(n3) process and is more expensive than the computation

39

0 € 0

I

* Figure 1. CPU Times for calculation of first derivatives of eigenvalLres and eigenvectors for a 60

x 60 matrix p. &

40

of the eigenvector itself which is an O(n2) process using the procedure

described in Section 3.1. This fact is significant as some authors have stated

the opposite[6,7].

41

3.5 Calculation of First and Second derivatives of

Eigenvalues only

Operation Count

7 2

-n3 + (K + 11rnn3 + I Adjoint Method

Direct Method

Di rect-Ad joi nt Met hod

I 2 + I (T) nq3K + 1) 3

/& + / ( y ) n 2 K + //T/n2(2K -k 1) 3

The Direct-Adjoint Method denotes the calculation of the eigenvector

derivatives by the Direct method and the eigenvalue second derivatives by the

Adjoint Method. The third term in the operation count for the Direct-Adjoint

Method is significant only when rn is small. From the operation count, it is

seen that the Direct-Adjoint Method is always cheaper than the Direct Method.

Hence, the choice lies between the Adjoint Method and the Direct-Adjoint

method. Here, considerations similar to those of the fast section hold and the

choice of method depends on the values of I and rn. When few eigenvalues

are of interest, the Direct-Adjoint method is cheaper. When many eigenvalues

are of interest, the Adjoint method is superior. But this advantage of APjoint 8

I

42

220 f ::r/ 160

W 80$ I 20

/ 0

0 p" / , / ,

A'

/ 0

0 0

0 6 0

Figure 2. CPU Times for calculation of second derivatives of eigenvalues for n 60 x 60 matrix

43

a

. Method diminishes as the number of design variables increases. This is again

illustrated for a 60 x 60 full matrix (K = 1) in Figure 2 on page 43.

44

Chapter 4

Approximate Eigenvalues of Modified Systems

4.1 Introduction

The eigenvalue problem to be solved is

Au(k) = h(k)u(k) (4.1 .l)

and

where A is a general complex matrix of order n and l ~ ( ~ ) , dk) and dk) are the

k -th eigenvalue and right and left eigenvectors respectively.

45

The matrix A differs by a small amount from a nominal matrix A,. The

eigenvalues Lo and the right and left eigenvectors uo and vo are taken to be

known.

A = A0 + AA

and

(4.1.3)

(4.1.4)

(4.1 5)

Throughout this chapter, we will also assume that the left eigenvectors are

normalized such that

The eigenvalues h and the eigenvectors u and v can be written as

#) = Ab"' + A p )

(4.1.7)

Let pa0 be the vector of nominal design variables and Apa be a vector of

perturbations from the nominal design variables so that

46

(4.1.8)

and

(4.1 -9)

Approximate q u a n t i l a are denoted by the subscript a . For example, an

approximation for the eigenvalue k ( k ) is denoted by hak). Al l derivatives are

evaluated at the nominal design.

An exact relation exists between Ah, AA, Au ana r iv as ioiiows.

(4.1 . lo)

The object is to obtain without solving a full eigenvalue problem. i,(k)

can be obtained exactly using eq. (4.1.10), if Au and Av are known. Since ine

exact values Au and Av cannot be obtained without solving a full eigenvalue

problem, various approximations can be formed based on the above

expression.

The approximations are broadly classified into

1. Derivative based approximations

2. Rayleigh-Quotient based approximations

47

3. Trace-theorem based approximations

Q* * 4. Others

4.2 Derivative Based Approximations

Derivative based approximations are of special importance in optimization

problems because first derivatives are required anyway in most optimization

algorithms.

The most common of the derivative based approximations are based

directly on truncated Taylor series. We will consider Linear and Quadratic

approximations in this category. The enormous cost associated with the

computation of any higher derivatives with multiple design variables

effectively precludes the possibility of using higher order approximations

based on the Taylor series.

4.2.1 Linear Approximation(L1N)

This is the simplest approximation ,a be considered in this work. Linear

Approximation is obtained by truncating the Taylor series expansion for the

eigenvalue after two terms.

48

(4.2.1 )

A linear approximation is usually inadequate in terms of accuracy because

eigenvalues are often highly non-linear functions of design variables. The

linear approximation will be referred to herein as LIN approximation.

4.2.2 Quadratic Approximation(QUAD)

The Quadratic approximation is obtained by truncating the Taylor series

expansion for the eigenvalue after three terms.

(4.2.2)

The quadratic approximation can be quite expensive for large orders of the

matrix or large number of design variables. Miura and Schmit[l4] used a

simplified form of the quadratic approximation and found that, considering

global efficiency, the higher cost of the quadratic approximation can

sometimes offset the higher accuracy in an optimization problem. The

quadratic approximation will be referred to herein as QUAD approximation.

Because of these efficiency considerations, attempts were made to

improve the accuracy of the linear approximations through the use of

intermediate variables with respect to which the eigenvalues may be nearly

49

linear. However, Miura and Schmit[l4] concluded that such intermediate

variables cannot be found for a general structural problem.

The accuracy of linear approximations can also be improved in another

fashion. This is by introducing non-linearities without, however, introducing

the second derivatives, which are expensive to calculate.

4.2.3 Conservative Approximation

In optimization applications, it is often desired to have a conservative

approximation. For eigenvalue problems, this usually means underestimating

the eigenvalues. Starnes and Haftka[51] proposed a hybrid approximation,

which is a combination of a linear approximation and a reciprocal

approximation (linear in l/pcI ) such that it is the most conservative

combination of the two. Since this approximation is only applicable to real

quantities, it is applied to the real part and/or the imaginary part of the

eigenvalue as required in particular applications.

(4.2.3)

where

50

Y

Even though this approximation is not, in general, more accurate than the

linear approximation, it is popular because it is more conservative than the

linear approximation and it is also convex.

4.2.4 Generalized Inverse Power Approximation

Non-linearities can also be introduced into the linear approximation in a

more direct manner, as in the Generalized Inverse Power approximation,

described by Prasad[52,53].

The linear approximation is first reformulated as

and the function ya is chosen as

(4.2.4)

(4.2.5)

where r is any real number.

51

Prasad[52] also gave an alternate formulation for r = 0. The real number

r. is a controlling parameter for the approximation. There is no obvious choice

for r in a general problem.

4.2.5 Generalized Hybrid Approximation

Woo[54] combined the concepts of the conservative approximation of

Starnes and Haftka[51] and the Generalized Inverse Power approximation and

defined a Generalized Hybrid approximation as

where

(4.2.6)

g being a real number and h being a positive integer such that

g 2 0 and g - h I -1. The choice of g and h is again not obvious, though

the fact that larger values for g and h make the approximation more

conservative may provide some guideline. Since this approximation too is

only applicable to real quantities, it is applied to the real part and/or the

imaginary part of the eigenvalue as required in particular applications.

52

4.2.6 Reduction Method(RDN)

A different approach to derivative based approximations is based on the

Taylor series approximation to the eigenvectors. This consists of reducing the

original eigenvalue problem to a series of smaller order eigenproblems and is

inspired by Noor’s concept of global approximation vectors[55]. The concept

of global approximation vectors is extended here to general matrices.

Noor’s formulation is simplified by assuming that the eigenvectors can be

treated as linear functions of the design variables. For the sake of simplicity,

consider only one design variable. Let

and substituting eq. (4.2.7) in eq. (4.1.1), we get,

If the above equation is premultiplied by [vbk) v(!)]~, then we have

(4.2.7)

(4.2.8)

(4.2.9)

53

and 1

Eq. (4.2.9) is a 2 x 2 linear eigenvalue problem and can be solved almost

effortlessly. Of the two eigenvalues of eq. (4.2.9), the one closest to the linear

approximation is chosen. In the case of multiple design variables, the only

change needed is to replace v!? and u'k,' by the respective derivatives in the

direction of change in design. It can be proved that, if the eigenvectors are

linear functions of the design variable, eq. (4.2.9) gives an exact eigenvalue

when uo and vo are normalized such that their first derivatives are respectively

orthogonal to them. This approximation will be referred to herein as the RDN

approximation.

4.3 Rayleigh Quotient Based Approximations

The Rayleigh quotient for the general eigenproblem given by eqs. (4.1.1)

and (4.1.2) is defined as

(4.3.1)

When all the eigenvalues of matrix A are distinct, then the Rayleigh

quotient R ( x , y ) has a stationary value at x = dk) and y = dk) (the right and

left eigenvectors associated with the eigenvalue A(k) ) for k = 1,2, ... , N.

Further, this stationary value is equal to h(k)[56]. That is,

(4.3.2)

The approximations in this section are based on the above property. They

seek to approximate the eigenvectors dk) and dk) and use them in the

Rayleigh quotient to calculate the approximate eigenvalue.

4.3.1 Rayleigh Quotient with Nominal Eigenvectors(RAL1)

This approximation is obtained by simply using the nominal eigenvectors

in the Rayleigh quotient.

= ,,!iTAufj

Eq u iva I e nt I y ,

(4.3.4)

The second expression is cheaper to compute when AA is sparse. We will

refer to this approximation as the RALl approximation.

55

4.3.2 Rayleigh Quotient with Linearly Approximated

Eigenvectors(RAL2)

The left and right eigenvectors are approximated from those of the nominal

matrix using a 2-term Taylor series:

The compu

+ ? v(k,)Ap,, a = l

ation of the eigenvector derivatives u(g and is (

in Chapters 2 and 3.

(4.3.5)

iscussed

These linearly approximated eigenvectors are then used in a Rayleigh

quotient to generate an approximate eigenvalue. Our RAL2 approximation is

therefore given by

(4.3.6)

4.3.3 Rayleig h Quotient with Perturbed Eigenvectors(RAL3)

In this algorithm, the perturbation AA in the matrix is used to evaluate the

perturbations in the right and the left eigenvectors either by Adjoint or Direct

Method, which are analogous to the Adjoint and Direct methods described in

Chapter 2. In the Adjoint method, assuming that the eigenvectors are

normalized according to eq. (2.2.3) and (2.2.4), the perturbations in the right

and left eigenvectors, 6 d k ) and a d k ) respectively, are calculated as follows:

where

and

(4.3.8)

(4.3.9)

(4.3.1 0)

(4.3.1 1)

McCalley[57] used this approach for error analysis of real symmetric

eigenvalue problems. Chen and Wada[58,59] and Chen and Garba[GO]used an

equivalent formulation for real symmetric matrices and obtained eigenvalue

approximations for all eigenvalues of the matrix simultaneously. Faddeev and

Faddeeva[61] applied this approach to general matrices to improve the

57

accuracy of approximate eigenvalues. Meirovitch and Ryland[62] presented

an extension to second order perturbations for general matrices.

In the Direct method, assuming that the right eigenvectors are normalized

according to eq. (2.2.4), t h e perturbations in eigenvectors are calculated as

follows. In eq. (4.1 .l), substitute eq. (4.1.3) and

to get, after ignoring second order perturbation terms,

A ~ G ~ ( ~ ) + AAUL~) = h i k ) d k ) t 6~ ( k ) uo (4

(4.3.1 1)

(4.3.12)

Eq. (4.3.12) is identical to eq. (2.3.24) when the derivatives are replaced by

perturbations. Hence, the same solution methods described in Section 2.3 can

be used to solve eq. (4.3.12) for the perturbations in the eigenvector. Thus,

6 U K ) = 0 (4.3.1 3)

and

Bo&y = 6r

where

(4.3.1 4)

58

- ( k ) uO Im-th column deleted

F y = with m-th element deleted

r = - AAuo ( k ) (4.3.15)

The perhrbations in left eigenvec!ors x e cS?ained simi!ar!y using fnrward

substitution.

The perturbations in the right and left eigenvectors are used to

approximate the eigenvectors. Thus,

The approximate eigenvectors are then used in the Rayleigh quotient as

in eq. (4.3.6) to form an approximation to the eigenvalue. Romstad, et. al [63]

presented both the Adjoint method approach and an equivalent Direct method

approach of this approximation for real symmetric matrices. However, they

obtained different numerical results with the two approaches because of an

error in their expressions for eigenvector perturbation by Adjoint method.

4.3.4 Rayleigh Quotient with One-step Inverse Iteration(RAL4)

The Inverse Iteration method has been recognized as a powerful tool for

accurate computation of eigenvectors[47,64]. The unusual feature of the

Inverse Iteration method is that accurate eigenvectors can be computed even

when the eigenvalue is not known accurately as long as the eigenvalue is

close enough to the correct eigenvalue. This feature can be used effectively

to improve the accuracy of a rapid but rough approximation. In addition, a

one-step inverse iteration is usually sufficient because most of the

improvement in accuracy normally occurs in the first step and in a

modification problem, the nominal eigenvector is available and provides an

excellent i nit ia I iterate.

In this algorithm, a first approximation hLk] to the eigenvalue is formed

as a Rayleigh Quotient (RALl) given by eq. (4.3.3). This approximation is then

used in a one-step inverse iteration scheme to obtain approximate

eigenvectors as follows:

(4.3.17)

The approximate eigenvectors are then used in the Rayleigh quotient as

in eq. (4.3.6) to form an approximation to the eigenvalue. We will refer to this

approximation as RAL4.

Note that the evaluation of the uLk) and vkk) requires only one matrix

factorization, since the second part of eq. (4.3.17) can be solved by forward

substitution using the same factored matrix used in solving the first part.

4.4 Trace-Theorem Based Approximations

These appr=xima?igns aye h-rcld nn i a , o I l I t n n s a n n i t o r - + i n r n mnthndr fnr uuoeu V I 1 V V b I I I \ I I U V V I I I I e I U t I v e IIIblIIUU3 IUI

finding the roots of a polynomial. We apply these methods to the

characteristic polynomial, p ( h ) of matrix A using only one step of the iteration

for the approximation.

The remarkable feature of the approximations in this section lies in the fact

that the coefficients of the characteristic polynomial need not be calculated

explicitly. This is achieved by employing the following result, known as the

Trace Theorem[56].

Trace Theorem: If p(h) Z 0, then

(4.4.1)

dP where p’(h) = -. d h

61

4.4.1 One-step Newton-Raphson Iteration(NRT1)

Here, we employ the Newton-Raphson formula

(4.4.2)

where hak) is an initial approximation for the eigenvalue

Using the Trace Theorem, eq. (4.4.1), an approximation is formulated as

(4.4.3)

Note that this approximation requires a matrix inversion. The initial

approximation hLk] is chosen as the Rayleigh quotient of eq. (4.3.3).

We will refer to this approximation as NRTl approximation.

4.4.2 Refined One-step Newton-Raphson Iteration(NRT2)

In this approximation, we utilize the second derivative of p ( h ) to obtain a

better approximation. To refine the Newton-Raphson Iteration using the

second derivative, consider the truncated Taylor series expansion

Use eq. (4.4.2) to eliminate hak) in the third term on the right hand side, to get

so that t h e refined Newton-Raphson formula is

From the definition of f (h ) , we have

p”(h) = p ( h ) [ f(lW) + f2(A) J

Now, differentiating eq. (4.4.1) with respect to I,, while noting that

d -1 2 - ( A - L l ) - ’ = [(A - hl) dh 1

we get,

- 1 2 f ( h ) = - Trace of [ (A - hl) ]

(3.4.5)

(4.4.6)

(4.3.73

(4.3.8)

(4.4.9)

This expression is then used in the refined Newton-Raphson formula of eq.

(4.4.6) to obtain our next approximation given below.

(4.4.1 0)

63

Note that, in evaluating f ( h ) using eq. (4.4.9), the matrix multiplication need

not be performed completely, since only the diagonal elements of the matrix

product are needed.

This approximation will be referred to herein as the NRT2 approximation.

4.4.3 One-step Laguerre Iteration(L1T)

Laguerre iteration is often used to compute eigenvalues and is known to

have excellent convergence properties[47]. For our purposes, we need only

one step of the Laguerre iteration. The one-step Laguerre iteration consists

of

This approximation also utilizes the second derivative of p ( L ) . The

derivation of eq. (4.4.11) is given in Wilkinson[47]. To obtain our

approximation, we rewrite the above, using the trace theorem, as

The evaluation of f ( h ) is described in Section 4.4.2. The sign in the

denominator is chosen so as to make the denominator have the greater

absolute value. We will refer to this approximation as the LIT approximation.

4.5 Other Approximations

An alternative approach to approximate eigenvalues is taken by Paipetis

and Croustalis[65] who developed an algorithm to approximate the coefficients

of the characteristic polynomial which is then solved to obtain approximate

eigenvalues. In addition to the numerical difficulties associated with t h e

evaluation of the coefficients of the characteristic polynomial and its solution,

this method severely restricts the characteristics of the system rnatrix.

We will consider one more approximation called [l ,l]Pade apprcximation.

4.5.1 [I ,I] Pade Approximation(PAD1)

We derive the [I ,I] Pade approximation by geometrical constructlon along

the lines of Johnson[66]. Let us for the moment assume that tile eibci-ivLAati.,

to be approximated is real.

Let hak] and A&) be the first and second approximations to the eigenvalue

The information contained in these approximations is exploited by

Aitken's method[67] to obtain a hopefully better approximation. We form t h e

differences

(4.5.1)

65

If Abk), hak) and h# is a converging series, we will have

We now draw a straight line through the points (Abk), Ehak?)) and (A&), 6hiv)

to extrapolate to 6h(k) = 0 on a (h(k), plot. This is illustrated in Figure 3.

It is expected that the extrapolated value hak) would be a better approximation

than either I$] or hky.

The result is

(4.5.3)

Although the motivation applied only to real eigenvalues, this result can

be immediately extended to complex eigenvalues. Using eqs.(4.5.1), eq. (4.5.3)

is rewritten as

(4.5.4)

If the approximations hak),h& are the linear and the quadratic

approximations, the approximation of eq. (4.5.4) can be recognized to be in the

form of the [1,1] Pade approximant. In the following, this is assumed and this

approximation will be referred to as PAD1 approximation.

66

Figure 3. Derivation of [I ,1] Pade Approximation using Geometric Coiistruction

67

Chapter 5

Accuracy and Efficiency of Eigenvalue

Approximations

5. I lntroduction

In Chapter 4, we listed several approximations for the eigenvalues of

general matrices. For a given application, the selection of the appropriate

approximation usually depends on the saving of computational time that a

given approximation entails. In the task of selecting a good approximation,

information about the accuracy and the efficiency of computation of the

approximations is essential. Accuracy and efficiency are not independent

elements in the selection of an approximation algorithm. Poor accuracy

usually translates into Ibw efficiency in the global process.

68

In this Chapter, the accuracy and efficiency considerations relating to the

approximations listed in the last chapter are discussed. The accuracy

considerations are treated in Section 5.2 and the efficiency considerations in

Section 5.3. The Conservative, Generalized Inverse Power and the

Generalized Hybrid approximations are not studied as their accuracy is

problem dependent and a general assessment is not feasible.

Some of the approximations discussed have been applied to symmetric

matrices by researchers in structural dynamics. However, there exists no

systematic comparison of accuracy and efficiency. To the best of the author’s

knowledge, the trace theorem based algorithms have never been used for

approximating eigenvalues of modified systems in the structural dynamics

I iterat ure.

5.2 Accuracy Considerations in Approximating

Eigenvalues

For simplicity of notation, we consider a single design variable. It will be

obvious, however, that the results of the error analysis are applicable to the

case of multiple design variables as well.

69

5.2.1 Order of an Approximation

70

For a useful comparison of the various approximations, we define the

order of an approximation as follows:

Definition: If an approximation ha to an eigenvalue h is such that the error

then that approximation is said to be of s-th order.

Note that, for a rigorous estimate of the error in an approximation,

information about both the order of the approximation as well as the

proportionality constant C in eq. (5.2.1) is needed. However, the order of the

approximation is usually the most important property of the approximation and

the proportionality constant is useful only when comparing approximations of

the same order. In this work, attention is focused on the order of the

approximation and the proportionality constant is discussed only in examples.

5.2.2 First Order Approximations

Among the approximations described in the last chapter, the linear

approximation(L1N) and the Rayleigh quotient with nominal eigenvectors

(RAL1) are first order approximations.

It is easy to show that the Linear approximation is of first order. Consider

the two term Taylor series for with remainder given by

(5.2.2)

where pa 5 5, pa + Ap, .

Comparing this to the linear approximation, eq. (4.2.1), we have the error

Hence the linear approximation is a first order approximation.

To find the error in the RALI approximation, subtract eq. (4.3.4) from the

exact expression of eq. (4.1.10). Ignoring the third order terms, we have

Considering that, to the first order,

AA, Au, Av = O(Ap,)

we have

(5.2.5)

71

Thus, RALl is a first order approximation.

5.2.3 Second Order Approximations

The Quadratic approximation and its improvement by [1,1] Pade

approximation(PAD1) are the second order approximations we described. To

show that the quadratic approximation is of second order, consider the three

term Taylor series with remainder given by

Comparing this to eq. (4.2.2), we have the error in the quadratic

approximation as

(5.2.7)

showing that the quadratic approximation is a second order approximation.

The PADl approximation is an improvement on the quadratic approximation

and so it is at least of second order.

The linear, quadratic and the PADl approximations are applicable to all

functions and do not take advantage of the special properties of the eigenvalue

problem. All the other approximations we are going to discuss are developed

72

specifically for approximating the eigenvalues and it will be shown that they

achieve higher accuracy with less computational effort.

5.2.4 Third Order Approximations

The reduction method RDN, the Rayleigh quotient based methods RAL2

and RAL3 and the one-step Newton-Raphson iteration NRTl are the third

srder me?hc!ds c!f apprr?xima?ior? !ha! we considered. ,As ?he first three

approximations RDN, RAL2 and RAL3 are closely related, we will derive the

order of only the RAL2 approximation and infer the orders of the RAL3 and the

RDN from this derivation.

Recall that in the RAL2 algorithm, the eigenvalue is approximated by using

linearly approximated left and right eigenvectors in the Rayleigh quotient.

Hence, we may write the two term Taylor series with remainder for the

eigenvectors as

(5.2.8)

Hence, the approximate eigenvectors can be rewritten in terms of the exact

eigenvectors as

73

Substituting these expressions in

get

(5.2.9)

eq. (4.3.6) and using the eqs. (4.1.1-2), we

where

(5.2.1 0)

(5.2.1 1)

Carrying out the long division and putting hak) = we find the error

in the RAL2 approximation as

(5 2.1 2)

It is hence proved that the RAL2 approximation is of third order.

It may be recalled that in the RAL3 approximation, we approximate the

eigenvalue by a Rayleigh quotient using left and right eigenvectors that were

obtained by using first order perturbations whereas in the RAL2 algorithm, we

approximated the eigenvectors using their first derivatives in a two-term

Taylor series. There is mathematically no difference between these two

methods and their difference lies only in the computational algorithms. It

follows then that RAL2 is also a third order approximation. There is indeed

little difference in the accuracy of the two approximations when they were

tested on example matrices.

The order of the reduction method approximation(RDN) is difficult to

obtain algebraically. However, we can infer the order of the RDN

approximation by comparing it to the RAL2 approximation. We first note that

the chief characteristic of the RAL2 approximation is that the eigenvector of

?he modified matrix is assumed to be in the subspace consisting only of the

linearly approximated eigenvector. The RDN approximation is more flexible

in that the eigenvector of the modified matrix is assumed to be in the subspace

consisting of the original eigenvector and its derivative. Thus, the RDN

approximation can be expected to be somewhat better than the RAL2

approximation. Hence, the RDN and the RAL2 approximations are expected

to be of the same order but possess a different proportionality constant in the

sense of eq. (5.2.1). This conclusion is validated by several numericai

experiments.

To derive the order of the one-step Newton-Raphson iteration(NRTl), we

first write the Taylor’s series for the characteristic polynomial, p(h) and its

derivative, p’(h), as

(5.2.13)

and

75

where primes denote derivatives evaluated at

Algorithm of eq. (4.4.2), modified by by subtracting

Then the Newton-Raphson

from both sides and

noting that ~ ( h ( ~ ) ) = 0,

(5.2.15)

may be written as

giving

(5.2.17)

We have chosen the initial approximation hak) to be the RALl approximation,

which has already been shown to be a first order approximation. Hence,

(A$) - = O(Ap:) (5.2.18)

so that

76

(5.2.19) (hELT1 - = O(AP,4)

establishing that the NRTl approximation is a third order approximation.

5.2.5 Higher Order Approximations

The remaining approximations, RAL4, NRT2 and LIT are all fifth order

approximations. We proceed to ob?ain the order of ?hese app:cxima?ims.

To get the order of the RAL4 approximation, we follow Ostrowski's

approach[68] using, however, the simplifying assumption that all the

eigenvalues are well-separated. Let U and V denote the matrices whose

columns are the right eigenvectors u and the left eigenvectors v respectively

of the matrix A. Let A denote the diagonal matrix of eigenvalues. From the

biorthogonal property of the left and right eigenvectors normalized as given

by eq. (2.2.3), we have

V T A U = A

Define

and T v U = l (5.2.20)

(5.2.21 )

From eqs. (5.2.20), we have

77

so that

(5.2.22)

(5.2.23)

Using the above and the definitions of eq. (5.2.21) after premultiplying the

first part of eq. (4.3.17) by VT, we obtain the relation

(5.2.24)

From the second part of eq. (4.3.17), we can obtain, in a similar manner,

Using eqs. (5.2.20-21), the RAL4 approximation can now be written as

(5.2.26)

(52.27)

78

Subtracting the exact eigenvalue

get after considerable algebra,

from both sides of this expression, we

(5.2.28)

_I :z 3 I k \ Then, if the eigenvaiues are weii separated aiiu II rbhy is a reasonably close

approximation of the eigenvalue

be negligible compared to the first. So, we may write

, the second term in the denominator will

(5.2.29)

However, hak] is taken to be the RALl approximation. Hence we have,

(5.2.30)

Subtracting from both sides as before,

79

(5.2.31)

(5.2.32)

When the eigenvalues are well-separated and hak) is close to the exact

eigenvalue the first term is dominant, so that

Substituting eq. (5.2.33) in eq. (5.2.29),

80

(5.2.33)

(5.2.34)

Now, putting hak) = h#AL4 and hak] = h#ALl, we have

Since RALl is a first order approximation as shown in Section 5.2.2,

(5.2.36)

proving that the RAL4 approximation is of fifth order.

The derivation of the order of the NRT2 approximation is analogous to that

of NRTl approximation given in the last section if eq. (4.4.10) is used in place

of eq. (4.4.2). After considerable algebra, we get

(5.2.37)

For the Laguerre Method, Wilkinson[47] gives

Since the initial approximation used in both these algorithms is of first order,

we have

81

- = O(Ap:) (5.2.39)

(5.2.40)

so that both the NRT2 and the LIT approximations are fifth order

approximations as stated.

5.2.6 Validation of the Theoretical Results

Taking logarithms of both sides of eq. (5.2.1), we have

log (A, - h) = (s + 1) log (Ap,) + constant (5.2.41 )

so that if the error in an s-th order approximation is plotted against the change

in design variable on log-log scale, one must obtain a straight line with slope

(s + 1).

The orders of approximations obtained in the last section are verified by

numerical experiments, summarized in the following figures. In order to

minimize the effect of round-off errors as much as possible, a small matrix of

order 5 is used for validation of theoretical results. The matrix elements are

the quadratic polynomials of a design variable where the coefficients are

generated using a random number generator. The errors in the approximate

eigenvalues are in comparison to the exact eigenvalue that is obtained by the

QR algorithm and improved by using the eigenvectors computed by inverse

82

1E-1

1 E - 2

1 E-3

1 E-4

1 E-5

1 E-6

1 E-7

1 E-8

1 E - 9

1E-10

1E-1 1

1 E - 1 2

1 E - 4 1 E - 3 1 E-2 1E-1

CHANGE I N D E S I G N V A R I A B L E - L I N - QUAD *+-e- P A D 1 - R A L l - R A L 2 - R A L 3 - R A L 4 - R D N - N R T l - N R T 2 e-+ L I T

Z

Figure 4. Relative Percentage errors in the Absolute value of the Eigenvalue 5 x 5 Matrix

83

1E-1

1 E-2

1 E-3

1 E-4

1 E-5

1 E-6

1 E-7

1 E-8

1 E-9

1E-10

1E-1 1

1E-12 -

Z

CHANGE I N D E S I G N V A R I A B L E

c--lclc L I N e-e-e QUAD P A D 1 .w--;n---n R A L I - R A L 2 - R A L 3

2--t--lt. R A L 4 - R O N N R T I - N R T 2 - L I T

Figure 5. Relative Percentage errors In the Real Part of the Eigenvalue 5 x 5 Matrix

84

1E-1

1 E-2

1 E-3

1 E-4

1 E-5

1 E-6

1 E-7

1 E - 8

1 E - 9

1 E - 1 0

1 E - 1 1

1 E - 1 2

1 E - 4 1 E-3 1 E - 2

CHANGE I N DESIGN V A R I A B L E

Z ICC--*- L I N QUAD P A D 1 - R A L I +-++- R A L 2 - R A L 3 - R A L 4 - RDN

c--*---*- N R T l - N R T 2 *+e- L I T

Figure 6. Relative Percentage errors in the Imaginary Part of the Eigenvalue 5 x 5 Matrix

85

iteration in a Rayleigh quotient. Figures 4, 5 and 6 show the errors in the

absolute value, the real part and in the imaginary part of an eigenvalue of the

5 x 5 matrix plotted on a log-log scale against the change in design variable.

In all the following figures, 10’ is represented in the FORTRAN notation 1Ei.

The slopes of the straight line segments in Figure 4, 5 and 6 agree very closely

with the orders of the approximations derived in Section 5.2. The difference

in accuracy between the high order and the low order approximations is

clear1 y apparent .

Approximations are also applied to one of the eigenvalues of a larger

matrix of order 40, generated in the same manner as the smaller matrix of

order 5 before. The results obtained are shown in Figures 7, 8 and 9. The

results for a flutter analysis matrix are shown in Figures 10, 11 and 12. The

generation of the flutter analysis matrix is described in Section 3.2. The design

variable used in the Figures 10, 11 and 12 is the reduced frequency,

k defined as (F) where o is the vibration frequency, b the blade

semi-chord and V the air speed in far field.

The deviations from straight line behavior appearing in Figures 10, 11 and

12 at small values of Apa are typical instances of round-off error with which

we are not concerned.

In all these Figures, approximations of equal order are indicated by

straight lines of equal slope. The proportionality constant, which reflects

accuracy, is indicated by the vertical position of the corresponding straight

line.

86

ai W

1E-1

1 E-2

1 E-3

1 E-4

1 E-5

1 E-6

1 E-7

1 E-8

1 E-9

1 E - 1 0

1E-1 1

1 E - 4 1 E - 3 1 E - 2 1 E - 1

CHANGE I N D E S I G N V A R I A B L E

-~c---+---lc L I N - QUAD - R A L l +-e-* P A D 1 - R A L 2 - R A L 3 - R A L 4 - RDN N R T I - N R T 2

++e- L I T

Z

Figure 7. Relative Percentage errors in the Absolute value of the Eigenvalue 40 x 40 Matrix

87

1E-1

1 E-2

1 E-3

1 E-4

1 E-5

1 E-6

1 E - 7

1 E-8

1 E - 9

1E-10

1E-I 1 1E-1 1 E-4 1 E - 3 1 E-2

C H A N G E I N DESIGN V A R I A B L E i*--*----*- L I N 8---e---f3 Q U A D *-e- P A D 1 - RAL1 +-+-+ R A L 2 - R A L 3 - R A L 4 +-A-& R O N - N R T I -I-+-+ N R T 2 .8----6---8 L I T

Z

Figure 8. Relative Percentage errors in the Real Part of the Eigenvalue 40 x 40 Matrix

1E-1

1 E-2

1 E-3

1 E-4

1 E-5

1 E-6

1 E-7

1 E-8

1 E-9

1 E - 1 0

1 E - I 1

1 E -4 1 E -3 1 E-2 1E-1 C H A N G E I N D E S I G N V A R I A B L E

Z -le--lc--r(c L I N QUAD 4t-e-e P A D 1 - R A L l - R A L 2 - R A L 3 - R A L 4 - RDN - N R T I - N R T 2

L I T

Figure 9. Relative Percentage errors in the Imaginary Part of the Eigenvnlue 40 x 40 Matrix

89

. . 1 E - 4 1 E - 3 1E-2 1E-1

CHANGE I N D E S I G N V A R I A B L E

Z -le*--+. L I N 8--e--8 QUAD - P A D 1 ~K---K--)c R A L l - R A L 2 -y--y---rc R A L 3 - R A L 4 - R O N +--+ N R T l - N R T 2 - L I T

Figure 10. Relative Percentage errors in the Absolute value of the Flutter Eigenvalue 40 x 40 Matrix

90

CHANGE I N DESIGN V A R I A B L E

Z -*--*--lc L I N P A D 1 - R A L 2 - R A L 4 - N R T l

e-e-e L I T

QUAD R A L 1 R A L 3 R O N N R T 2

Figure 11. Relative Percentage errors in the Real Part of the Flutter Eigenvalue 40 x 40 Matrix

'3. 91

1 E - 3

1 E-4

1E-5

1 E-6

1 E - 7

1 E - 8

1 E - 9

1 E - 1 0

1 E-4 1 E - 3 1E-2 1 E - 1

CHANGE I N D E S I G N V A R I A B L E

Z -IC--+-C L I N - Q U A D - P A D 1 - R A L l - R A L 2 - R A L 3 2--3--+- R A L 4 - RDN - N R T I - N R T 2 - L I T

Figure 12. Relative Percentage errors in the Imaginary Part of the Flutter Eigenvalue 40 x 40 Matrix

92

It is seen that both the first order approximations have about the same

accuracy and that the improvement achieved by the PAD1 approximation over

the quadratic approximation is marginal. The reduction method approximation

with good consistency shows substantially more accuracy than the other third

order methods. Among the fifth order approximations, the NRTl

approximation is consistently poorer in accuracy than others.

5.3 Efficiency Considerations in Approximating

Eigenvalues

Efficiency is probably the most important consideration in the comparative

evaluation of the various approximations, particularly in the context of design

optimization. The comparison is once again made in terms of variables which

significantiy influence the cost of computing an approximation. in addition io

the size of the matrix n, the number of design parameters rn and the number

of eigenvalues of interest I that we considered in Chapter 3, we have an

additional variable in the approximation context and this is the number of

design points d at which an approximation is sought based on the same

nominal design.

Tables 4 and 5 present the operation counts for all the approximations

studied. The operation counts include the necessary computations of the left

and right eigenvectors at the nominal design represented by the matrix A,, i f

93

Table 4. Operation Counts for First and Second Order Approximations

First 0 rd e r A p p rox i m a t i o n s

Approximation Operation Count

7 2

2

/(-n2 + Kmn2 i- dn) LIN

RALl /(-n2 7 + Kdn2)

Second Order Approximations

Approximat ion Operation Count

QUAD ln3 + I ( y ) n2K + / r n n 2 ( 2 ~ + 1) + d/n2 3

- Direct-Adjoint Method

( y ) n2K i- d/n2 -n3 + (K + 1 ) m d + / 7 2

- Adjoint Method

PAD1 Same as QUAD

Table 5. Operation Counts for Third and Higher Order Approximations

Approxi mat ion

RDN

RAL2

RAL3

N RT1

Approximation

RAL4

N RT2

LIT

Third Order Approximations

Operation Count

n

dln

Higher Order Approximations

Operation Count

Direct Method

Adjoint Method

Direct Method

. Adinin? . - , - . . . Methnd . . . - . . . -

Direct Method

Adjoint Method

95

they are significant. But they do not include the operations for the calculation

of the nominal eigenvalues ho since these do not affect the comparison of the

efficiency of the different approximations. When the derivatives of

eigenvectors or the second derivatives of the eigenvalues are needed for an

approximation, operations counts for both the Adjoint method algorithm and

Direct method algorithm are given separately. The details of the Adjoint and

the Direct methods are discussed in Chapters 2 and 3.

Note that the computational expense of RAL1, RAL3, RAL4 and all the fifth

order approximations is independent of the number of design variables.

5.4 Discussion of Approximations

5.4.1 Case When No Derivatives are Available

The results depicted in Figures 4-12 show that both the first order

approximations we studied, LIN and RALl , are practically identical in

accuracy. Comparing their operation counts, we conclude that, when first

order accuracy is acceptable, LIN approximation is preferable if the number

of design variables is small and the number of design points for approximation

is large. When the number of design variables is large, the RALl

approximation is more efficient.

96

The experience of numerical experiments also shows that the improvement

achieved by the PADl approximation over the quadratic approximation is

marginal so that we have two second order approximations of nearly the same

accuracy. The behavior of the [1,1] Pade approximation based on initial

approximations other than the linear and the quadratic has not been studied.

However, the operation counts show that the evaluation of the second

derivatives is very expensive. The third and higher order approximations are

not only more accurate but are also more efficient so that the second order

approximations QUAD and PADl can be completely dropped from

consideration. Among the third order methods, RAL2, RAL3 and NRTl have

similar or same accuracy while the Reduction method RDN shows higher

accuracy in most cases, sometimes close to that of some of the fifth order

approximations. This is explained by the fact that the reduction method

approximates the eigenvectors in a subspace of two vectors whereas RAL2

and RAL3 approximations use a subspace of only one vector. However, the

reduction method is also more expensive than RAL2 and RAL3 as shown by

the operation counts in Table 5.

Hence, among the third order methods, the trade-off between accuracy

and efficiency determines the choice of the approximation. When accuracy is

more important than efficiency, the reduction method is chosen over the

others. When efficiency is more important, we have the choice between RAL2

and RAL3. The Newton-Raphson approximation(NRT1) is dropped from

consideration because there are higher order methods which give higher

97

accuracy with the same computational expense. Note that the Direct method

computation of the RAL3 approximation is more expensive than the Adjoint

method computation except for very few eigenvalues of interest and very few

design points of approximation. The choice between RAL2 and RAL3 is similar

to that between LIN and RALl in the first order case. RAL2 approximation is

preferable if the number of design variables is small and the number of design

points for approximation is large. When the number of design variables is

large, the RAL3 approximation is more efficient.

The higher order approximations are particularly efficient when the

number of design variables is large and the number of design points and the

number of eigenvalues of interest is small. The RAL4 algorithm is somewhat

more efficient than the other two. Among these approximations, for the cases

tested, the RAL4 and the LIT methods give better accuracy than NRT2. As the

NRT2 approximation is no cheaper than the LIT approximation, this eliminates

the NRT2 approximation from consideration. Whether there is any

considerable difference in accuracy between the RAL4 and the LIT methods

requires further investigation. In the absence of any such difference, the RAL4

approximation may be considered to be the best higher order approximation

available in terms of accuracy and efficiency.

The computational cost of the higher order approximations escalates

rapidly to equal that of the exact computation of eigenvalues. Comparing the

operation count of the RAL4 approximation and the exact computation, we

98

note that the RAL4 approximation is more expensive than the exact

computation when the product dl i s above 30.

5.4.2 Case When Derivatives are Available

Most design optimization algorithms require first derivatives of

constraints. Design optimization of dynamic response in structures almost

always involves constraints on the eigenvalues and sometimes constraints on

eigenvectors are also involved. In such cases, the first derivatives of the

eigenvalues and perhaps eigenvectors are already available free for use in

approximations so that the operation counts for derivative-based

approximations given in Tables 4 and 5 are reduced, affecting some of the

conclusions. In this section, we discuss the relative merits of the

approximations when the first derivatives are already available.

We first consider the case when constraints are placed only on the

eigenvalues so that the first derivatives of only eigenvalues are available free.

In such a case, the operation count for only the linear approximation is

affected and is shown in Table 6. The linear approximation is now an O ( n )

process and is practically free in terms of computational expense compared to

any other approximation. Thus, when constraints are placed on the

derivatives of eigenvalues, the linear approximation is the best approximation

unless particularly high accuracy is required. If high accuracy is desired, the

conclusions of Section 5.3.1 still hold.

99

Table 6. Operation Counts for First Order Approximations when Eigenvalue Derivatives are Free

Approxi mation

LIN

RALl

First Order Approximations

Operation Count

Idn

100

We next consider the case when the constraints are placed on both the

eigenvalues and eigenvectors so that all first derivatives are available free. In

such a case, the operation counts for all derivative-based approximations are

affected and are shown in Tables 7 and 8.

The derivative based approximations except the quadratic approximation

are much more attractive when al l the first derivatives are available. The

linear approximation is again practically free so that it always makes sense to

use linear approximation in terms of efficiency. The third order

approximations RDN and RAL2 are now O(n2) processes and are substantially

cheaper than the fifth order approximations which are still O ( n 3 ) processes.

However, the RDN approximation is now twice as expensive as the RAL2

approximation and the additional computational expense of the RDN

approximation is less easily justified even though it is somewhat more

accurate. The quadratic approximation is again more expensive than the more

accurate third order approximations. Operation counts for the fifth order

approximations are not affected by the availability of the first derivatives and

hence the conclusions regarding the same are also unaffected.

101

Table 7. Operation Counts for First and Second Order Approximations when All First Derivatives are Free

First Order Approximations

Approximation

LIN

RALl

Operation Count

Idn

7 2

Id(-n2 + Kn2)

Approxi mat ion

QUAD

Second Order Approximations

Operation Count

I (T) n2K + lmn2K + dln2

- Direct-Adjoint Method

(K + 1)mn3 + / (y)n2r + dln2

- Adjoint Method

PAD1 Same as QUAD

Table 8. Operation Counts for Third and Higher Order Approximations when All First Derivatives are Free

Third Order Approximations

Approxi mati on

RDN

RAL2

RAL3

NRTl

Approxi mat ion

RAL4

N RT2

LIT

Operation Count

4dln

dln

7 2 -n3 + dln2(K t 3)

din

Higher Order Approximations

Operation Count

dln

dln

Adjoint Method

103

Chapter 6

Conclusions

The large computational expense associated with the flutter optimization

of a cascade of rotating blades motivated this study. The problem of

computational expense is attacked from two fronts, sensitivity analysis and

approximations. The existing literature was surveyed in both fields and

improvements are suggested. General recommendations for the selection of

the most efficient algorithms are presented.

The normalization of the eigenvector needs to be properly related to its

derivative. In practice, this means that the derivative of the eigenvector is to

be normalized before it is used, to conform to the normalization of the

eigenvector itself. When the eigenvector is not normalized in a unique

manner, its derivative cannot be evaluated. It has been shown that fixing one

of the components of the eigenvector is the best normalizing condition for

computation of the derivative.

104

In the sensitivity analysis part, the algorithms presently available for

computing exactly the derivatives of eigenvalues and eigenvectors are

classified into Adjoint and Direct Methods. Adjoint Methods use both the left

and the right eigenvectors whereas the Direct Methods use only the right

eigenvectors. The Adjoint Methods and the Direct Methods found in the

literature are extended to apply to eigenvectors normalized in the manner

described above. Algorithms that compute approximate derivatives are not

studied as their implementation is problem dependent or complicated.

The Adjoint and the Direct methods are examined for their efficiency under

different sets of conditions. The choice reflects whether the solution of the

adjoint problem is worth the extra computational expense. The solution of the

adjoint problem is shown to be cheaper than the solution of the direct problem

because left and right eigenvectors can be calculated using the same factored

matrix. It is concluded that if only the first derivatives of eigenvalues are

required, the solution of the adjoint problem is worth the expense since the

Adjoint method is superior to the Direct Method. When first derivatives of

eigenvectors are also required, the decision is dependent on the problem size,

the number of design variables and the number of eigenvalues of interest. The

Direct method is more competitive if the number of design variables is large

and the eigenvalues of interest are few. When the first and second derivatives

of eigenvalues are required, similar considerations hold. It is also shown that

once the first derivatives of eigenvectors are calculated, the second

105

derivatives of eigenvalues are calculated more efficiently by the Adjoint

method than by the Direct method.

In the approximation part, many existing approximation methods are

applied to general matrices. Some new approximation methods, inspired by

the computational techniques in linear algebra, are proposed. Noor’s concept

of global approximation vectors is simplified and then extended to general

matrices to arrive at another approximation called the reduction method.

The approximations are classified as Derivative Based, Rayleigh Quotient

Based, Trace Theorem Based and [l ,l]Pade approximations according to their

theoretical origin. In terms of accuracy, the approximations are reclassified

according to the order of the error expected from the approximation. The

approximation methods are also examined for computational expense as this

is a significant issue in the selection of a particular method. At each order of

accuracy, the approximations are compared in terms of their efficiency and

general recommendations are made. Additional recommendations are made

for the case when the derivatives are already available. In particular, it is

concluded that the quadratic approximation is inferior to many other

approximations both in accuracy and efficiency.

The analysis performed in this work is applicable also to the generalized

eigenvalue problem in a straightforward manner so that all the conclusions are

valid for the usual structural stability problem. However, the conclusions are

limited by the assumption of distinct and well-separated eigenvalues of

interest. The sensitivity analysis and approximation methods for multiple or

106

closely spaced eigenvalues is fraught with difficulties, numerical as well as

theoretical. The multiple eigenvalue case is suggested as a topic for further

research.

107

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1. Report No.

NASA CR-179538

S e n s i t i v i t y A n a l y s i s and Approx imat ion Methods f o r General E igenvalue Problems

~ ~~~~

2. Government Accession No.

7. Author@)

Durbha V. Murthy and Raphael T. H a f t k a

3 Security Classif (of this report) U n c l a s s i f i e d

9 Performing Organization Name and Address

V i r g i n i a Po ly techn ic I n s t i t u t e and S ta te U n i v e r s i t y Aerospace and Ocean Eng ineer ing Blacksburg, V i r g i n i a 24061

-_ 20 Security Classif (of this page) 21 No of pages 22 Price'

U n c l a s s i f i e d 123 A06

2. Sponsoring Agency Name and Address

N a t i o n a l Aeronaut ics and Space A d m l n i s t r a t i o n Washington, D.C. 20546

_ _ _ _ _ _ ~ 5. Supplementary Notes

3. Recipient's Catalog No.

5. Report Date

October 1986

6. Performing Organization Code

8. Performing Organization Report No.

None

10. Work Unit No.

505-63- 11

11. Contract or Grant No.

NAG3 - 347

13. Type of Report and Period Covered

C o n t r a c t o r Report F i n a l

14. Sponsoring Agency Code

P r o j e c t Manager, K r i shna Rao V . Kaza, S t ruc tu res D i v i s i o n , NASA I.ewis Research Center.

6. Abstract

O p t i m i z a t i o n o f dynamic systems i n v o l v i n g complex n o n - h e r m i t i a n m a t r i c e s i s o f t e n c o m p u t a t i o n a l l y expensive. Major c o n t r i b u t o r s t o t h e computat ional expense a r e t h e s e n s i t i v i t y a n a l y s i s and r e a n a l y s i s o f a m o d i f i e d des ign . The p resen t work seeks t o a l l e v i a t e t h i s computat ional burden by i d e n t i f y i n g e f f i c i e n t s e n s i t i v i t y a n a l y s i s and approximate r e a n a l y s i s methods. For t h e a lgebra ic . e igenva lue prob lem i n v o l v i n g non-he rm i t i an ma t r i ces , a lgo r i t hms f o r s e n s i t i v i t y a n a l y s i s and approx imate r e a n a l y s i s a r e c l a s s i f i e d , compared and eva lua ted f o r e f f i c i e n c y and accuracy. Proper e igenvec to r n o r m a l i z a t i o n i s d iscussed. An improved method f o r c a l c u l a t i n g d e r i v a t i v e s o f e igenvec to rs i s proposed bdsed on a more r a t i o n a l n o r m a l i z a t i o n c o n d i t i o n and t a k i n g advantage o f m a t r i x s p a r s i t y . numer ica l aspects o f t h i s method a r e a lso d iscussed. To a l l e v i a t e t h e problem o f r e a n a l y s i s , va r ious approx ima t ion methods f o r e igenvalues a r e proposed and eva lua ted . s e r i e s . Severa l app rox ima t ion methods are developed based on t h e g e n e r a l i z e d Ray le igh q u o t i e n t f o r t h e e igenvalue problem. Approx imat ion methods based on t r a c e theorem g i v e h i g h accuracy w i t h o u t needing any d e r i v a t i v e s . Operat ion counts f o r t h e computat ion o f t h e approximat ions a r e g i ven . General recommenda t i o n s a r e made f o r t h e s e l e c t i o n o f a p p r o p r i a t e app rox ima t ion technique as a f u n c t i o n o f t h e m a t r i x s i z e , number o f design v a r i a b l e s , number o f e igenvalues o f i n t e r e s t and t h e number of des ign p o i n t s a t which approx ima t ion i s sought.

Impor tan t

L i n e a r and q u a d r a t i c approximat ions a r e based d i r e c t l y on t h e Tay lo r

7 Kay Words (Suggested by Author@))

S e n s i t i v i t y ; Eigenvalue; Approx imat ion; O p t i m i z a t i o n

-~ - -

U n c l a s s i f i e d u n l i m i t e d SIAH Category 39

'For sale by the National Technical Information Service. Springfield, Virginia 22161